Finite Group Databasebeta

← #127855

Subgroups — #127855

Order 512.

815 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20898C211yesmaximal, characteristic
2#17364C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#7285C214yesmaximal, complemented
5C23×M16C411yescharacteristic
6C22×C4×D4C2211yescharacteristic
7#2920—21no—
8C2×#407—21no—
9C2×#407C411yescharacteristic
10C22×C22⋊C8—21no—
11#1870C412yes—
12#1870C2212yes—
13#1869—21no—
14#1869—21no—
15#1728—21no—
16#1466—21no—
17#1466C412yes—
18#1462—21no—
19#1462C412yes—
20#1460—21no—
21#1460C2212yes—
22#1459—21no—
23#1459—21no—
24#1459C412yes—
25#1459C2212yes—
26C2×#324C412yes—
27C2×#323—21no—
28C23×D4Q811yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—44no—
34C22×M16—41no—
35C22×M16—41no—
36C22×M16C2×C412yes—
37C23×C8—21no—
38C23×C8—41no—
39C2×C4×D4—41no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4—41no—
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4Q811yescharacteristic
50C2×C4×D4C2311yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42D411yescharacteristic
55#432—22no—
56#432—41no—
57#431—22no—
58#431—41no—
59#423—41no—
60#407—22no—
61#407—44no—
62#407—41no—
63#407—41no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—41no—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8C2×C412yes—
74C4×M16—41no—
75#336—22no—
76#336—22no—
77#324—22no—
78#324C2×C414yes—
79#323—22no—
80#323—22no—
81#323—22no—
82#323—22no—
83#323—22no—
84#323C2×C414yes—
85C25C4⋊C411yescharacteristic
86C25C4⋊C411yescharacteristic
87C22×D4—24no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4—42no—
91C22×D4C4⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C22×D4C2×Q811yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4C4211yescharacteristic
108C23×C4C4211yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—24no—
121C2×M16—44no—
122C2×M16—42no—
123C2×M16—48no—
124C2×M16—44no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—44no—
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—41no—
132C2×M16—41no—
133C2×M16—42no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—44no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—82no—
142C22×C8—82no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154C4×D4—41no—
155C4×D4—41no—
156C4×D4—41no—
157C4×D4—41no—
158C4×D4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C4211yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C4⋊C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4C4211yescharacteristic
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4C4⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C22×C411yescharacteristic
176C2×C22⋊C4C2×D411yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C4⋊C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C4⋊C8—42no—
186C4⋊C8—42no—
187C22⋊C8—24no—
188C22⋊C8—24no—
189C22⋊C8—44no—
190C22⋊C8—44no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197#97—42no—
198C4×C8—42no—
199C24—21no—
200C24—22no—
201C24—21no—
202C24—22no—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—41no—
207C24—42no—
208C24—42no—
209C24—81no—
210C24—81no—
211C24#9511yescharacteristic
212C24#9511yescharacteristic
213C24#9511yescharacteristic
214C24C2×C4⋊C411yescharacteristic
215C24C2×C4⋊C411yescharacteristic
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—48no—
228C2×D4—48no—
229C2×D4#10814yes—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—81no—
285C22×C4#9511yescharacteristic
286C22×C4#9511yescharacteristic
287C22×C4#9511yescharacteristic
288C22×C4#9511yescharacteristic
289C22×C4#9511yescharacteristic
290C22×C4#9911yescharacteristic
291C22×C4#9911yescharacteristic
292C22×C4#9911yescharacteristic
293C22×C4#9911yescharacteristic
294C22×C4C2×C4211yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299M16—44no—
300M16—44no—
301M16—44no—
302M16—44no—
303M16—82no—
304M16—82no—
305M16—84no—
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—48no—
309C2×C8—42no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—84no—
316C2×C8—82no—
317C2×C8—82no—
318C2×C8—84no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C23—24no—
340C23—24no—
341C23—22no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—84no—
373C23—84no—
374C23—84no—
375C23—84no—
376C23#34112yes—
377C23#34112yes—
378C23#34211yescharacteristic
379C23#34211yescharacteristic
380C23C2×#9511yescharacteristic
381C23#43012yes—
382D4—44no—
383D4—44no—
384D4—44no—
385D4—44no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4#34211yescharacteristic
439C2×C4#34211yescharacteristic
440C2×C4AΣL(1,𝔽16)12yes—
441C2×C4#35212yes—
442C2×C4C2×#9911yescharacteristic
443C2×C4C2×#9911yescharacteristic
444C2×C4C2×#10812yes—
445C8—44no—
446C8—82no—
447C8—84no—
448C8—84no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—48no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—84no—
478C22—81no—
479C22—81no—
480C22—84no—
481C22—84no—
482C22#139012yes—
483C22C2×#34111yescharacteristic
484C22#173811yescharacteristic
485C22C2×#34211yescharacteristic
486C4—21no—
487C4—21no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4#139012yes—
497C4C2×AΣL(1,𝔽16)11yescharacteristic
498C4C2×#35211yescharacteristic
499C2—22no—
500C2—21no—
501C2—21no—
502C2—41no—
503C2—42no—
504C2—42no—
505C2—42no—
506C2—81no—
507C2#805711yescharacteristic
508C2C2×#139011yescharacteristic
509C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7285C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#208982
#173642
C2×#14592
#72852